Forecasting from different time horizon

Question

I have a time series of log returns (from 01-01-2007 to 31-12-2012). I need to predict the t+1,t+2, ....... t+1510 log stock returns. Basically, what I need is to predict the 1 day ahead return for every single day. So I would like to predict the return on the 02-01-2007, the 03-01-2007........till the 01-01-2013.

I have therefore done the following coding using fGARCH in R:

      AIG <- garchFit(formula = ~arma(1, 1) + garch(1, 1), data =R.AXP, cond.dist = "std")
      Predict_AIG <- predict(AIG, n.ahead = 1510)

But this gives me the prediction on a single forecast horizon (so it gives me the forecast 1, 2,3.....1510 days from the 01-01-2007) not the forecast on the 02-01-2007, 03-01-2007,...........01-01-2013.

Please can anyone help me to find a different way of coding. Thank you.

Answer 1

• I get the same values(for the mean forecast) up to the last date (row 1510). Am I doing something wrong?

An $h$-step forecast from an AR(1) model is $$\hat{x}_{t+h}=\mu+\phi^h (x_t-\mu)$$ where $\phi$ is the AR(1) coefficient and $\mu$ is the mean of the series.

1. You may be getting the same values for all but the first forecast (which is the only one affected by the MA(1) term) if the estimated AR(1) coefficient $\hat{\phi}$ is equal to zero.

2. You may be getting approximately equal values for all but the first few forecasts if the estimated AR(1) coefficient $\phi$ is close to zero. For $h$ sufficiently large $\phi^h$ will be close to zero and you will not see any differences between forecasts due to rounding.

   • What values should I consider as the predicted returns?

meanForecast seems to be the column of predicted returns.

• What does the mean error mean?

Not sure. Perhaps it indicates the width of the prediction interval: meanForecast$\pm$ meanError should give you a ?% prediction interval (the exposition here may help a bit).

• How is the mean error of any use to me?

If you only care about point forecasts, you may neglect the mean error for the moment. But typically you are also interested in how precise your point forecast is likely to be. Then you look at prediction intervals to get the idea where the point forecast should lie with ?% certainty.

Finally, are you sure you want to forecast as far as 1510 periods ahead? Your forecast could be reliable one or a few periods ahead but not a hundred or a thousand periods ahead. By the time of 1510 periods your data generating process (DGP) may change a lot.

An alternative to making just one point forecast is to simulate your process many times (at least a thousand) and look at the realizations. By using some summary statistics over the realizations you could perhaps get a broader view of what may happen in the future. Unfortunately, this does not account for the possible evolution of your DGP over time, which may be a problem.